Virginia Department of Education s7

Superintendent of Public Instruction Patricia I. Wright, Ed.D.

Assistant Superintendent for Instruction Linda M. Wallinger, Ph.D.

Office of Elementary Instruction Mark R. Allan, Ph.D., Director Deborah P. Wickham, Ph.D., Mathematics Specialist

Office of Middle and High School Instruction Michael F. Bolling, Mathematics Coordinator

Acknowledgements The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D. who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework.

NOTICE The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in employment or in its educational programs or services.

The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s Web site at http://www.doe.virginia.gov. Virginia Mathematics Standards of Learning Curriculum Framework 2009 Introduction

The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity the content that all teachers should teach and all students should learn.

Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for each standard. The Curriculum Framework is divided into three columns: Understanding the Standard; Essential Understandings; and Essential Knowledge and Skills. The purpose of each column is explained below.

Understanding the Standard This section includes background information for the teacher (K-8). It contains content that may extend the teachers’ knowledge of the standard beyond the current grade level. This section may also contain suggestions and resources that will help teachers plan lessons focusing on the standard.

Essential Understandings This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the Standards of Learning. In Grades 6-8, these essential understandings are presented as questions to facilitate teacher planning.

Essential Knowledge and Skills Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills that define the standard.

The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills from Standards of Learning presented in previous grades as they build mathematical expertise. FOCUS 6–8 STRAND: NUMBER AND NUMBER SENSE GRADE LEVEL 6

In the middle grades, the focus of mathematics learning is to  build on students’ concrete reasoning experiences developed in the elementary grades;  construct a more advanced understanding of mathematics through active learning experiences;  develop deep mathematical understandings required for success in abstract learning experiences; and  apply mathematics as a tool in solving practical problems.

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students in the middle grades focus on mastering rational numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to most middle school mathematics topics.  Students develop an understanding of integers and rational numbers by using concrete, pictorial, and abstract representations. They learn how to use equivalent representations of fractions, decimals, and percents and recognize the advantages and disadvantages of each type of representation. Flexible thinking about rational number representations is encouraged when students solve problems.  Students develop an understanding of the properties of operations on real numbers through experiences with rational numbers and by applying the order of operations.  Students use a variety of concrete, pictorial, and abstract representations to develop proportional reasoning skills. Ratios and proportions are a major focus of mathematics learning in the middle grades.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 1 STANDARD 6.1 STRAND: NUMBER AND NUMBER SENSE GRADE LEVEL 6

6.1 The student will describe and compare data, using ratios, and will use appropriate notations, such as , a to b, and a:b.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  A ratio is a comparison of any two quantities. A  What is a ratio? The student will use problem solving, mathematical ratio is used to represent relationships within and A ratio is a comparison of any two quantities. A ratio is communication, mathematical reasoning, connections, between sets. used to represent relationships within a set and between and representations to two sets. A ratio can be written using fraction form  Describe a relationship within a set by comparing  A ratio can compare part of a set to the entire set ( ), a colon (2:3), or the word to (2 to 3). (part-whole comparison). part of the set to the entire set.  A ratio can compare part of a set to another part  Describe a relationship between two sets by of the same set (part-part comparison). comparing part of one set to a corresponding part of the other set.  A ratio can compare part of a set to a corresponding part of another set (part-part  Describe a relationship between two sets by comparison). comparing all of one set to all of the other set.  A ratio can compare all of a set to all of another  Describe a relationship within a set by comparing set (whole-whole comparison). one part of the set to another part of the same set.  The order of the quantities in a ratio is directly  Represent a relationship in words that makes a related to the order of the quantities expressed in the a comparison by using the notations , a:b, and a to relationship. For example, if asked for the ratio of the b number of cats to dogs in a park, the ratio must be b. expressed as the number of cats to the number of dogs, in that order.  Create a relationship in words for a given ratio expressed symbolically.  A ratio is a multiplicative comparison of two numbers, measures, or quantities.  All fractions are ratios and vice versa.  Ratios may or may not be written in simplest form.  Ratios can compare two parts of a whole.  Rates can be expressed as ratios.

6.2 The student will a) investigate and describe fractions, decimals and percents as ratios; b) identify a given fraction, decimal or percent from a representation; c) demonstrate equivalent relationships among fractions, decimals, and percents; and d) compare and order fractions, decimals, and percents.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Percent means “per 100” or how many “out of  What is the relationship among fractions, decimals The student will use problem solving, mathematical 100”; percent is another name for hundredths. and percents? communication, mathematical reasoning, Fractions, decimals, and percents are three different connections, and representations to  A number followed by a percent symbol (%) is ways to express the same number. A ratio can be written  Identify the decimal and percent equivalents for equivalent to that number with a denominator of 100 using fraction form ( ), a colon (2:3), or the word to (2 to numbers written in fraction form including repeating (e.g., 30% = = = 0.3). 3). Any number that can be written as a fraction can be decimals.  Percents can be expressed as fractions with a expressed as a terminating or repeating decimal or a denominator of 100 (e.g., 75% = = ). percent.  Represent fractions, decimals, and percents on a number line.  Percents can be expressed as decimal (e.g., 38% = = 0.38).  Describe orally and in writing the equivalent relationships among decimals, percents, and fractions  Some fractions can be rewritten as equivalent that have denominators that are factors of 100. fractions with denominators of powers of 10, and can be represented as decimals or percents  Represent, by shading a grid, a fraction, decimal, 6 and percent. (e.g., = 10 = = 0.60 = 60%).  Represent in fraction, decimal, and percent form a  Decimals, fractions, and percents can be represented given shaded region of a grid. using concrete materials (e.g., Base-10 blocks, number lines, decimal squares, or grid paper).  Compare two decimals through thousandths using manipulatives, pictorial representations, number lines,  Percents can be represented by drawing shaded and symbols (<, ,, >, =). regions on grids or by finding a location on number lines.  Compare two fractions with denominators of 12 or less using manipulatives, pictorial representations,  Percents are used in real life for taxes, sales, data ,, description, and data comparison. number lines, and symbols (<, >, =).  Fractions, decimals and percents are equivalent  Compare two percents using pictorial forms representing a given number. representations and symbols (<, ,, >, =).  The decimal point is a symbol that separates the  Order no more than 3 fractions, decimals, and Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 3 STANDARD 6.2 STRAND: NUMBER AND NUMBER SENSE GRADE LEVEL 6

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only) whole number part from the fractional part of a number. percents (decimals through thousandths, fractions with denominators of 12 or less), in ascending or descending  The decimal point separates the whole number order. amount from the part of a number that is less than one.  The symbol can be used in Grade 6 in place of “x” to indicate multiplication. 1  Strategies using 0, and 1 as benchmarks can be 2 used to compare fractions. 1  When comparing two fractions, use as a 2 4 3 benchmark. Example: Which is greater, or ? 7 9 4 1 is greater than because 4, the numerator, 7 2 represents more than half of 7, the denominator. The denominator tells the number of parts that make the 3 1 whole. is less than because 3, the numerator, 9 2 is less than half of 9, the denominator, which tells the number of parts that make the whole. Therefore, 4 3 > . 7 9  When comparing two fractions close to 1, use distance from 1 as your benchmark. Example: Which is 6 8 6 1 8 1 greater, or ? is away from 1 whole. is 7 9 7 7 9 9 Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 4 STANDARD 6.2 STRAND: NUMBER AND NUMBER SENSE GRADE LEVEL 6

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only) 1 1 6 away from 1 whole. Since > , then is a greater 7 9 7 8 8 6 distance away from 1 whole than so > . 9 9 7  Students should have experience with fractions such 1 as , whose decimal representation is a terminating 8 1 2 decimal (e. g., = 0.125) and with fractions such as , 8 9 whose decimal representation does not end but continues 2 to repeat (e. g., = 0.222…). The repeating decimal 9 can be written with ellipses (three dots) as in 0.222… or denoted with a bar above the digits that repeat as in 0.2 .

6.3 The student will a) identify and represent integers; b) order and compare integers; and c) identify and describe absolute value of integers.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Integers are the set of whole numbers, their  What role do negative integers play in practical The student will use problem solving, mathematical opposites, and zero. situations? communication, mathematical reasoning, Some examples of the use of negative integers are connections, and representations to  Positive integers are greater than zero. found in temperature (below 0), finance (owing  Identify an integer represented by a point on a money), below sea level. There are many other  Negative integers are less than zero. number line. examples.  Zero is an integer that is neither positive nor  Represent integers on a number line. negative.  How does the absolute value of an integer compare to the absolute value of its opposite?  Order and compare integers using a number line.  A negative integer is always less than a positive They are the same because an integer and its integer. opposite are the same distance from zero on a  Compare integers, using mathematical symbols (<, >, =).  When comparing two negative integers, the negative number line. integer that is closer to zero is greater.  Identify and describe the absolute value of an integer.  An integer and its opposite are the same distance from zero on a number line. For example, the opposite of 3 is -3.  The absolute value of a number is the distance of a number from zero on the number line regardless of direction. Absolute value is represented as -6 = 6.

 On a conventional number line, a smaller number is always located to the left of a larger number (e.g., –7 lies to the left of –3, thus –7 < –3; 5 lies to the left of 8 thus 5 is less than 8).

6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Using manipulatives to build conceptual  When multiplying fractions, what is the meaning of The student will use problem solving, mathematical understanding and using pictures and sketches to link the operation? communication, mathematical reasoning, concrete examples to the symbolic enhance students’ When multiplying a whole by a fraction such as 3 x connections, and representations to understanding of operations with fractions and help 1  Demonstrate multiplication and division of fractions students connect the meaning of whole number , the meaning is the same as with multiplication of 2 using multiple representations. computation to fraction computation. 1 whole numbers: 3 groups the size of of the whole.  Model algorithms for multiplying and dividing with  Multiplication and division of fractions can be 2 fractions using appropriate representations. represented with arrays, paper folding, repeated addition, When multiplying a fraction by a fraction such as repeated subtraction, fraction strips, pattern blocks and 2 3 area models. , we are asking for part of a part. 3 4  When multiplying a whole by a fraction such as When multiplying a fraction by a whole number such 1 1 3 x , the meaning is the same as with multiplication of as x 6, we are trying to find a part of the whole. 2 2 1 whole numbers: 3 groups the size of of the whole.  What does it mean to divide with fractions? 2 For measurement division, the divisor is the number of  When multiplying a fraction by a fraction such as groups and the quotient will be the number of groups in the dividend. Division of fractions can be explained as 2 3 , we are asking for part of a part. how many of a given divisor are needed to equal the 3 4 1 2 given dividend. In other words, for the question  When multiplying a fraction by a whole number 4 3 1 2 1 such as x 6, we are trying to find a part of the whole. is, “How many make ?” 2 3 4 For partition division the divisor is the size of the  For measurement division, the divisor is the number group, so the quotient answers the question, “How of groups. You want to know how many are in each of much is the whole?” or “How much for one?” those groups. Division of fractions can be explained as how many of a given divisor are needed to equal the 1 2 given dividend. In other words, for , the question 4 3 2 1 is, “How many make ?” 3 4  For partition division the divisor is the size of the Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 7 STANDARD 6.4 STRAND: NUMBER AND NUMBER SENSE GRADE LEVEL 6

6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only) group, so the quotient answers the question, “How much is the whole?” or “How much for one?”

6.5 The student will investigate and describe concepts of positive exponents and perfect squares.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  In exponential notation, the base is the number that  What does exponential form represent? The student will use problem solving, mathematical is multiplied, and the exponent represents the number of Exponential form is a short way to write repeated communication, mathematical reasoning, times the base is used as a factor. In 83, 8 is the base and multiplication of a common factor such as connections, and representations to 3 is the exponent. 4 5 x 5 x 5 x 5 = 5 .  Recognize and describe patterns with exponents that  A power of a number represents repeated  What is the relationship between perfect squares and are natural numbers, by using a calculator. multiplication of the number by itself a geometric square?  Recognize and describe patterns of perfect squares (e.g., 83 = 8  8  8 and is read “8 to the third power”). A perfect square is the area of a geometric square not to exceed 20 2 , by using grid paper, square tiles, whose side length is a whole number.  Any real number other than zero raised to the zero tables, and calculators. power is 1. Zero to the zero power (0) is undefined.  Recognize powers of ten by examining patterns in a  Perfect squares are the numbers that result from place value chart: 104 = 10,000, 103 = 1000, 102 = 100, multiplying any whole number by itself 101 = 10, 10 0 =1. (e.g., 36 = 6  6 = 6 2 ).  Perfect squares can be represented geometrically as the areas of squares the length of whose sides are whole numbers (e.g., 1  1, 2  2, or 3  3). This can be modeled with grid paper, tiles, geoboards and virtual manipulatives.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 9 FOCUS 6–8 STRAND: COMPUTATION AND ESTIMATION GRADE LEVEL 6

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students develop conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring meaning to why procedures work and make sense.  Students develop and refine estimation strategies and develop an understanding of when to use algorithms and when to use calculators. Students learn when exact answers are appropriate and when, as in many life experiences, estimates are equally appropriate.  Students learn to make sense of the mathematical tools they use by making valid judgments of the reasonableness of answers.  Students reinforce skills with operations with whole numbers, fractions, and decimals through problem solving and application activities.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 10 STANDARD 6.6 STRAND: COMPUTATION AND ESTIMATION GRADE LEVEL 6

6.6 The student will a) multiply and divide fractions and mixed numbers; and b) estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Simplifying fractions to simplest form assists with  How are multiplication and division of fractions and The student will use problem solving, mathematical uniformity of answers. multiplication and division of whole numbers alike? communication, mathematical reasoning, Fraction computation can be approached in the same connections, and representations to  Addition and subtraction are inverse operations as way as whole number computation, applying those  Multiply and divide with fractions and mixed are multiplication and division. concepts to fractional parts. numbers. Answers are expressed in simplest form.  It is helpful to use estimation to develop  What is the role of estimation in solving problems?  Solve single-step and multistep practical problems computational strategies. For example, Estimation helps determine the reasonableness of 7 3 that involve addition and subtraction with fractions and 2 is about of 3, so the answer is between 2 and answers. 8 4 mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. 3. Answers are expressed in simplest form.  When multiplying a whole by a fraction such as  Solve single-step and multistep practical problems 1 3 , the meaning is the same as with multiplication of that involve multiplication and division with fractions 2 and mixed numbers that include denominators of 12 or 1 less. Answers are expressed in simplest form. whole numbers: 3 groups the size of of the whole. 2

 When multiplying a fraction by a fraction such as 2 3 , we are asking for part of a part. 3 4  When multiplying a fraction by a whole number 1 such as 6 , we are trying to find a part of the whole. 2

6.7 The student will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Different strategies can be used to estimate the  What is the role of estimation in solving problems? The student will use problem solving, mathematical result of computations and judge the reasonableness of Estimation gives a reasonable solution to a problem communication, mathematical reasoning, the result. For example: What is an approximate answer when an exact answer is not required. If an exact answer connections, and representations to for 2.19  0.8? The answer is around 2 because 2  1 = is required, estimation allows you to know if the  Solve single-step and multistep practical problems 2. calculated answer is reasonable. involving addition, subtraction, multiplication and  Understanding the placement of the decimal point is division with decimals expressed to thousandths with no very important when finding quotients of decimals. more than two operations. Examining patterns with successive decimals provides meaning, such as dividing the dividend by 6, by 0.6, by 0.06, and by 0.006.  Solving multistep problems in the context of real- life situations enhances interconnectedness and proficiency with estimation strategies.  Examples of practical situations solved by using estimation strategies include shopping for groceries, buying school supplies, budgeting an allowance, deciding what time to leave for school or the movies, and sharing a pizza or the prize money from a contest.

6.8 The student will evaluate whole number numerical expressions, using the order of operations.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  The order of operations is a convention that defines  What is the significance of the order of operations? The student will use problem solving, mathematical the computation order to follow in simplifying an The order of operations prescribes the order to use to communication, mathematical reasoning, expression. simplify expressions containing more than one connections, and representations to operation. It ensures that there is only one correct  Simplify expressions by using the order of  The order of operations is as follows: answer. – First, complete all operations within grouping operations in a demonstrated step-by-step approach. The symbols*. If there are grouping symbols within other expressions should be limited to positive values and not grouping symbols, do the innermost operation first. include braces { } or absolute value | |. – Second, evaluate all exponential expressions.  Find the value of numerical expressions, using order – Third, multiply and/or divide in order from left of operations, mental mathematics, and appropriate to right. tools. Exponents are limited to positive values. – Fourth, add and/or subtract in order from left to right. * Parentheses ( ), brackets [ ], braces {}, and the division 3+ 4 bar – as in should be treated as grouping symbols. 5+ 6  The power of a number represents repeated multiplication of the number (e.g., 83 = 8 · 8 · 8). The base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In the example, 8 is the base, and 3 is the exponent.  Any number, except 0, raised to the zero power is 1. Zero to the zero power is undefined.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 13 FOCUS 6–8 STRAND: MEASUREMENT GRADE LEVEL 6

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students develop the measurement skills that provide a natural context and connection among many mathematics concepts. Estimation skills are developed in determining length, weight/mass, liquid volume/capacity, and angle measure. Measurement is an essential part of mathematical explorations throughout the school year.  Students continue to focus on experiences in which they measure objects physically and develop a deep understanding of the concepts and processes of measurement. Physical experiences in measuring various objects and quantities promote the long-term retention and understanding of measurement. Actual measurement activities are used to determine length, weight/mass, and liquid volume/capacity.  Students examine perimeter, area, and volume, using concrete materials and practical situations. Students focus their study of surface area and volume on rectangular prisms, cylinders, pyramids, and cones.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 14 STANDARD 6.9 STRAND: MEASUREMENT GRADE LEVEL 6

6.9 The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Making sense of various units of measure is an  What is the difference between weight and mass? The student will use problem solving, mathematical essential life skill, requiring reasonable estimates of Weight and mass are different. Mass is the amount of communication, mathematical reasoning, what measurements mean, particularly in relation to matter in an object. Weight is the pull of gravity on the connections, and representations to other units of measure. mass of an object. The mass of an object remains the  Estimate the conversion of units of length, – 1 inch is about 2.5 centimeters. same regardless of its location. The weight of an object weight/mass, volume, and temperature between the U.S. – 1 foot is about 30 centimeters. changes dependent on the gravitational pull at its Customary system and the metric system by using – 1 meter is a little longer than a yard, or about 40 location. ballpark comparisons. inches. Ex: 1 L 1qt. Ex: 4L 4 qts. – 1 mile is slightly farther than 1.5 kilometers.  How do you determine which units to use at different times? Units – 1 kilometer is slightly farther than half a mile.  Estimate measurements by comparing the object to of measure are determined by the attributes of the object – 1 ounce is about 28 grams. be measured against a benchmark. – 1 nickel has the mass of about 5 grams. being measured. Measures of length are expressed in – 1 kilogram is a little more than 2 pounds. linear units, measures of area are expressed in square – 1 quart is a little less than 1 liter. units, and measures of volume are expressed in cubic – 1 liter is a little more than 1 quart. units. – Water freezes at 0°C and 32°F.  Why are there two different measurement systems? – Water boils at 100°C and 212°F. Measurement systems are conventions invented by – Normal body temperature is about 37°C and different cultures to meet their needs. The U.S. 98°F. Customary System is the preferred method in the – Room temperature is about 20°C and 70°F. United States. The metric system is the preferred  Mass is the amount of matter in an object. Weight is system worldwide. the pull of gravity on the mass of an object. The mass of an object remains the same regardless of its location. The weight of an object changes dependent on the gravitational pull at its location. In everyday life, most people are actually interested in determining an object’s mass, although they use the term weight, as shown by the questions: “How much does it weigh?” versus “What is its mass?”  The degree of accuracy of measurement required is determined by the situation.

6.9 The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Whether to use an underestimate or an overestimate is determined by the situation.  Physically measuring objects along with using visual and symbolic representations improves student understanding of both the concepts and processes of measurement.

6.10 The student will a) define pi (π) as the ratio of the circumference of a circle to its diameter; b) solve practical problems involving circumference and area of a circle, given the diameter or radius; c) solve practical problems involving area and perimeter; and d) describe and determine the volume and surface area of a rectangular prism.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Experiences in deriving the formulas for area,  What is the relationship between the circumference The student will use problem solving, mathematical perimeter, and volume using manipulatives such as tiles, and diameter of a circle? communication, mathematical reasoning, one-inch cubes, adding machine tape, graph paper, The circumference of a circle is about 3 times the connections, and representations to geoboards, or tracing paper, promote an understanding measure of the diameter.  Derive an approximation for pi (3.14 or ) by of the formulas and facility in their use.†  What is the difference between area and perimeter? gathering data and comparing the circumference to the  The perimeter of a polygon is the measure of the Perimeter is the distance around the outside of a figure diameter of various circles, using concrete materials or distance around the polygon. while area is the measure of the amount of space computer models.  Circumference is the distance around or perimeter enclosed by the perimeter.  Find the circumference of a circle by substituting a of a circle.  What is the relationship between area and surface value for the diameter or the radius into the formula C = d or C = 2r.  The area of a closed curve is the number of area? Surface nonoverlapping square units required to fill the region area is calculated for a three-dimensional figure. It is the  Find the area of a circle by using the formula sum of the areas of the two-dimensional surfaces that enclosed by the curve. A = r2. make up the three-dimensional figure.  The perimeter of a square whose side measures s is  Apply formulas to solve practical problems 4 times s (P = 4s), and its area is side times side (A = involving area and perimeter of triangles and rectangles. s2).  Create and solve problems that involve finding the  The perimeter of a rectangle is the sum of twice the circumference and area of a circle when given the length and twice the width [P = 2l + 2w, or diameter or radius. P = 2(l + w)], and its area is the product of the length and the width (A = lw).  Solve problems that require finding the surface area of a rectangular prism, given a diagram of the prism with  The value of pi () is the ratio of the circumference the necessary dimensions labeled. of a circle to its diameter.  Solve problems that require finding the volume of a  The ratio of the circumference to the diameter of a rectangular prism given a diagram of the prism with the circle is a constant value, pi (), which can be necessary dimensions labeled. approximated by measuring various sizes of circles.

6.10 The student will a) define pi (π) as the ratio of the circumference of a circle to its diameter; b) solve practical problems involving circumference and area of a circle, given the diameter or radius; c) solve practical problems involving area and perimeter; and d) describe and determine the volume and surface area of a rectangular prism.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  The fractional approximation of pi generally used is .  The decimal approximation of pi generally used is 3.14.  The circumference of a circle is computed using C= p d or C= 2p r , where d is the diameter and r is the radius of the circle.  The area of a circle is computed using the formula A= p r 2 , where r is the radius of the circle.  The surface area of a rectangular prism is the sum of the areas of all six faces ( SA=2 lw + 2 lh + 2 wh ).  The volume of a rectangular prism is computed by multiplying the area of the base, B, (length x width) by the height of the prism (V= lwh = Bh ).

†Revised March 2011

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 18 FOCUS 6–8 STRAND: GEOMETRY GRADE LEVEL 6

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students expand the informal experiences they have had with geometry in the elementary grades and develop a solid foundation for the exploration of geometry in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.  Students learn geometric relationships by visualizing, comparing, constructing, sketching, measuring, transforming, and classifying geometric figures. A variety of tools such as geoboards, pattern blocks, dot paper, patty paper, miras, and geometry software provides experiences that help students discover geometric concepts. Students describe, classify, and compare plane and solid figures according to their attributes. They develop and extend understanding of geometric transformations in the coordinate plane.  Students apply their understanding of perimeter and area from the elementary grades in order to build conceptual understanding of the surface area and volume of prisms, cylinders, pyramids, and cones. They use visualization, measurement, and proportional reasoning skills to develop an understanding of the effect of scale change on distance, area, and volume. They develop and reinforce proportional reasoning skills through the study of similar figures.  Students explore and develop an understanding of the Pythagorean Theorem. Mastery of the use of the Pythagorean Theorem has far-reaching impact on subsequent mathematics learning and life experiences.

The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student experiences that should lead to conceptual growth and understanding.  Level 0: Pre-recognition. Geometric figures are not recognized. For example, students cannot differentiate between three-sided and four-sided polygons.  Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look somewhat the same. (This is the expected level of student performance during grades K and 1.)  Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures. (Students are expected to transition to this level during grades 2 and 3.)

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 19 FOCUS 6–8 STRAND: GEOMETRY GRADE LEVEL 6

 Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusions are understood, but the role and significance of deduction is not understood. (Students should transition to this level during grades 5 and 6 and fully attain it before taking algebra.)  Level 4: Deduction. Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. Students should be able to supply reasons for steps in a proof. (Students should transition to this level before taking geometry.)

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 20 STANDARD 6.11 STRAND: GEOMETRY GRADE LEVEL 6

6.11 The student will a) identify the coordinates of a point in a coordinate plane; and b) graph ordered pairs in a coordinate plane.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  In a coordinate plane, the coordinates of a point are  Can any given point be represented by more than The student will use problem solving, mathematical typically represented by the ordered pair (x, y), where x one ordered pair? communication, mathematical reasoning, is the first coordinate and y is the second coordinate. The coordinates of a point define its unique location in a connections, and representations to However, any letters may be used to label the axes and coordinate plane. Any given point is defined by only one  Identify and label the axes of a coordinate plane. the corresponding ordered pairs. ordered pair.  Identify and label the quadrants of a coordinate  The quadrants of a coordinate plane are the four  In naming a point in the plane, does the order of the plane. regions created by the two intersecting perpendicular two coordinates matter? Yes. number lines. Quadrants are named in counterclockwise The first coordinate tells the location of the point to the  Identify the quadrant or the axis on which a point is order. The signs on the ordered pairs for quadrant I are left or right of the y-axis and the second point tells the positioned by examining the coordinates (ordered pair) (+,+); for quadrant II, (–,+); for quadrant III, (–, –); and location of the point above or below the x-axis. Point (0, of the point. for quadrant IV, (+,–). 0) is at the origin.  Graph ordered pairs in the four quadrants and on the  In a coordinate plane, the origin is the point at the axes of a coordinate plane. intersection of the x-axis and y-axis; the coordinates of this point are (0,0).  Identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane.  For all points on the x-axis, the y-coordinate is 0. For all points on the y-axis, the x-coordinate is 0.  Relate the coordinate of a point to the distance from each axis and relate the coordinates of a single point to  The coordinates may be used to name the point. (e.g., another point on the same horizontal or vertical line.† the point (2,7)). It is not necessary to say “the point whose coordinates are (2,7)”.

†Revised March 2011

6.12 The student will determine congruence of segments, angles, and polygons.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Congruent figures have exactly the same size and  Given two congruent figures, what inferences can be The student will use problem solving, mathematical the same shape. drawn about how the figures are related? communication, mathematical reasoning, The congruent figures will have exactly the same connections, and representations to  Noncongruent figures may have the same shape but size and shape. not the same size.  Characterize polygons as congruent and noncongruent according to the measures of their sides @  Given two congruent polygons, what inferences can  The symbol for congruency is . and angles. be drawn about how the polygons are related?  The corresponding angles of congruent polygons Corresponding angles of congruent polygons will  Determine the congruence of segments, angles, and have the same measure, and the corresponding sides of have the same measure. Corresponding sides of polygons given their attributes. congruent polygons have the same measure. congruent polygons will have the same measure.  Draw polygons in the coordinate plane given  The determination of the congruence or coordinates for the vertices; use coordinates to find the noncongruence of two figures can be accomplished by length of a side joining points with the same first placing one figure on top of the other or by comparing coordinate or the same second coordinate. Apply these the measurements of all sides and angles. techniques in the context of solving practical and †  Construction of congruent line segments, angles, mathematical problems. and polygons helps students understand congruency. †Revised March 2011

6.13 The student will describe and identify properties of quadrilaterals.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  A quadrilateral is a closed planar (two-dimensional)  Can a figure belong to more than one subset of The student will use problem solving, mathematical figure with four sides that are line segments. quadrilaterals? communication, mathematical reasoning, Any figure that has the attributes of more than one connections, and representations to  A parallelogram is a quadrilateral whose opposite subset of quadrilaterals can belong to more than one  Sort and classify polygons as quadrilaterals, sides are parallel and opposite angles are congruent. subset. For example, rectangles have opposite sides parallelograms, rectangles, trapezoids, kites, rhombi, and of equal length. Squares have all 4 sides of equal  A rectangle is a parallelogram with four right squares based on their properties. Properties include length thereby meeting the attributes of both subsets. angles. number of parallel sides, angle measures and number of  Rectangles have special characteristics (such as congruent sides. diagonals are bisectors) that are true for any rectangle.  Identify the sum of the measures of the angles of a  To bisect means to divide into two equal parts. quadrilateral as 360°.  A square is a rectangle with four congruent sides or a rhombus with four right angles.  A rhombus is a parallelogram with four congruent sides.  A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the nonparallel sides are called legs. If the legs have the same length, then the trapezoid is an isosceles trapezoid.  A kite is a quadrilateral with two pairs of adjacent congruent sides. One pair of opposite angles is congruent.  Quadrilaterals can be sorted according to common attributes, using a variety of materials.  Quadrilaterals can be classified by the number of parallel sides: a parallelogram, rectangle, rhombus, and square each have two pairs of parallel sides; a trapezoid has only one pair of parallel sides; other quadrilaterals have no parallel sides.  Quadrilaterals can be classified by the measures of Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 23 STANDARD 6.13 STRAND: GEOMETRY GRADE LEVEL 6

6.13 The student will describe and identify properties of quadrilaterals.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only) their angles: a rectangle has four 90° angles; a trapezoid may have zero or two 90° angles.  Quadrilaterals can be classified by the number of congruent sides: a rhombus has four congruent sides; a square, which is a rhombus with four right angles, also has four congruent sides; a parallelogram and a rectangle each have two pairs of congruent sides.  A square is a special type of both a rectangle and a rhombus, which are special types of parallelograms, which are special types of quadrilaterals.  The sum of the measures of the angles of a quadrilateral is 360°.  A chart, graphic organizer, or Venn Diagram can be made to organize quadrilaterals according to attributes such as sides and/or angles.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 24 FOCUS 6–8 STRAND: PROBABILITY AND STATISTICS GRADE LEVEL 6

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students develop an awareness of the power of data analysis and probability by building on their natural curiosity about data and making predictions.  Students explore methods of data collection and use technology to represent data with various types of graphs. They learn that different types of graphs represent different types of data effectively. They use measures of center and dispersion to analyze and interpret data.  Students integrate their understanding of rational numbers and proportional reasoning into the study of statistics and probability.  Students explore experimental and theoretical probability through experiments and simulations by using concrete, active learning activities.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 25 STANDARD 6.14 STRAND: PROBABILITY AND STATISTICS GRADE LEVEL 6

6.14 The student, given a problem situation, will a) construct circle graphs; b) draw conclusions and make predictions, using circle graphs; and c) compare and contrast graphs that present information from the same data set.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  To collect data for any problem situation, an  What types of data are best presented in a circle The student will use problem solving, mathematical experiment can be designed, a survey can be conducted, graph? communication, mathematical reasoning, or other data-gathering strategies can be used. The data Circle graphs are best used for data showing a connections, and representations to can be organized, displayed, analyzed, and interpreted to relationship of the parts to the whole.  Collect, organize and display data in circle graphs answer the problem. by depicting information as fractional.  Different types of graphs are used to display different  Draw conclusions and make predictions about data types of data. presented in a circle graph. – Bar graphs use categorical (discrete) data (e.g., months or eye color).  Compare and contrast data presented in a circle – Line graphs use continuous data (e.g., graph with the same data represented in other graphical temperature and time). forms. – Circle graphs show a relationship of the parts to a whole.  All graphs include a title, and data categories should have labels.  A scale should be chosen that is appropriate for the data.  A key is essential to explain how to read the graph.  A title is essential to explain what the graph represents.  Data are analyzed by describing the various features and elements of a graph.

6.15 The student will a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Measures of center are types of averages for a data  What does the phrase “measure of center” mean? The student will use problem solving, mathematical set. They represent numbers that describe a data set. This is a collective term for the 3 types of averages communication, mathematical reasoning, Mean, median, and mode are measures of center that are for a set of data – mean, median, and mode. connections, and representations to useful for describing the average for different situations.  Find the mean for a set of data. – Mean works well for sets of data with no very  What is meant by mean as balance point? Mean can be defined as the point on a number line where high or low numbers.  Describe the three measures of center and a situation the data distribution is balanced. This means that the – Median is a good choice when data sets have a in which each would best represent a set of data. couple of values much higher or lower than most of the sum of the distances from the mean of all the points others. above the mean is equal to the sum of the distances of all  Identify and draw a number line that demonstrates – Mode is a good descriptor to use when the set of the data points below the mean. This is the concept of the concept of mean as balance point for a set of data. data has some identical values or when data are not mean as the balance point. conducive to computation of other measures of central tendency, as when working with data in a yes or no survey.  The mean is the numerical average of the data set and is found by adding the numbers in the data set together and dividing the sum by the number of data pieces in the set.  In grade 5 mathematics, mean is defined as fair- share.  Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean. This is the concept of mean as the balance point.  Defining mean as balance point is a prerequisite for understanding standard deviation.

 The median is the middle value of a data set in Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 27 STANDARD 6.15 STRAND: PROBABILITY AND STATISTICS GRADE LEVEL 6

6.15 The student will a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only) ranked order. If there are an odd number of pieces of data, the median is the middle value in ranked order. If there is an even number of pieces of data, the median is the numerical average of the two middle values.  The mode is the piece of data that occurs most frequently. If no value occurs more often than any other, there is no mode. If there is more than one value that occurs most often, all these most-frequently-occurring values are modes. When there are exactly two modes, the data set is bimodal.

6.16 The student will a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  The probability of an event occurring is equal to the  How can you determine if a situation involves The student will use problem solving, mathematical ratio of desired outcomes to the total number of possible dependent or independent events? Events are communication, mathematical reasoning, outcomes (sample space). independent when the outcome of one has no effect on connections, and representations to the outcome of the other. Events are dependent when the  Determine whether two events are dependent or  The probability of an event occurring can be outcome of one event is influenced by the outcome of independent. represented as a ratio or the equivalent fraction, decimal, the other. or percent.  Compare and contrast dependent and independent  The probability of an event occurring is a ratio events. between 0 and 1.  Determine the probability of two dependent events. – A probability of 0 means the event will never occur.  Determine the probability of two independent – A probability of 1 means the event will always events. occur.  A simple event is one event (e.g., pulling one sock out of a drawer and examining the probability of getting one color).  Events are independent when the outcome of one has no effect on the outcome of the other. For example, rolling a number cube and flipping a coin are independent events.  The probability of two independent events is found by using the following formula: P( Aand B )= P ( A ) P ( B ) Ex: When rolling two number cubes simultaneously, what is the probability of rolling a 3 on one cube and a 4 on the other? 1 1 1 P(3 and 4)= P (3)� P (4) � 6 6 36

6.16 The student will a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Events are dependent when the outcome of one event is influenced by the outcome of the other. For example, when drawing two marbles from a bag, not replacing the first after it is drawn affects the outcome of the second draw.  The probability of two dependent events is found by using the following formula: P( Aand B )= P ( A ) P ( B after A ) Ex: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the bag on the first pick and then without replacing the blue ball in the bag, picking a red ball on the second pick? 1 1 1 P(blue and red)= P (blue)� P (red after blue) � 3 2 6

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 30 FOCUS 6–8 STRAND: PATTERNS, FUNCTIONS, AND ALGEBRA GRADE LEVEL 6

Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands.  Students extend their knowledge of patterns developed in the elementary grades and through life experiences by investigating and describing functional relationships.  Students learn to use algebraic concepts and terms appropriately. These concepts and terms include variable, term, coefficient, exponent, expression, equation, inequality, domain, and range. Developing a beginning knowledge of algebra is a major focus of mathematics learning in the middle grades.  Students learn to solve equations by using concrete materials. They expand their skills from one-step to two-step equations and inequalities.  Students learn to represent relations by using ordered pairs, tables, rules, and graphs. Graphing in the coordinate plane linear equations in two variables is a focus of the study of functions.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 31 STANDARD 6.17 STRAND: PATTERNS, FUNCTIONS, AND ALGEBRA GRADE LEVEL 6

6.17 The student will identify and extend geometric and arithmetic sequences.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Numerical patterns may include linear and  What is the difference between an arithmetic and a The student will use problem solving, mathematical exponential growth, perfect squares, triangular and other geometric sequence? communication, mathematical reasoning, polygonal numbers, or Fibonacci numbers. While both are numerical patterns, arithmetic connections, and representations to sequences are additive and geometric sequences are  Investigate and apply strategies to recognize and  Arithmetic and geometric sequences are types of multiplicative. numerical patterns. describe the change between terms in arithmetic patterns.  In the numerical pattern of an arithmetic sequence,  Investigate and apply strategies to recognize and students must determine the difference, called the describe geometric patterns. common difference, between each succeeding number in  Describe verbally and in writing the relationships order to determine what is added to each previous between consecutive terms in an arithmetic or geometric number to obtain the next number. Sample numerical sequence. patterns are 6, 9, 12, 15, 18, ; and 5, 7, 9, 11, 13, .  Extend and apply arithmetic and geometric  In geometric number patterns, students must sequences to similar situations. determine what each number is multiplied by to obtain the next number in the geometric sequence. This  Extend arithmetic and geometric sequences in a multiplier is called the common ratio. Sample geometric table by using a given rule or mathematical relationship. number patterns include 2, 4, 8, 16, 32, …; 1, 5, 25, 125, 625, …; and 80, 20, 5, 1.25, …  Compare and contrast arithmetic and geometric sequences.  Strategies to recognize and describe the differences between terms in numerical patterns include, but are not  Identify the common difference for a given limited to, examining the change between consecutive arithmetic sequence. terms, and finding common factors. An example is the  Identify the common ratio for a given geometric pattern 1, 2, 4, 7, 11, 16, sequence.

6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  A one-step linear equation is an equation that  When solving an equation, why is it necessary to The student will use problem solving, mathematical requires one operation to solve. perform the same operation on both sides of an equal communication, mathematical reasoning, connections sign? and representation to  A mathematical expression contains a variable or a combination of variables, numbers, and/or operation To maintain equality, an operation performed on  Represent and solve a one-step equation, using a symbols and represents a mathematical relationship. An one side of an equation must be performed on the variety of concrete materials such as colored chips, expression cannot be solved. other side. algebra tiles, or weights on a balance scale.  A term is a number, variable, product, or quotient in  Solve a one-step equation by demonstrating the an expression of sums and/or differences. In 7x2 + 5x – steps algebraically. 3, there are three terms, 7x2, 5x, and 3.  Identify and use the following algebraic terms  A coefficient is the numerical factor in a term. For appropriately: equation, variable, expression, term, and example, in the term 3xy2, 3 is the coefficient; in the term coefficient. z, 1 is the coefficient.  Positive rational solutions are limited to whole numbers and positive fractions and decimals.  An equation is a mathematical sentence stating that two expressions are equal.  A variable is a symbol (placeholder) used to represent an unspecified member of a set.

6.19 The student will investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Identity elements are numbers that combine with  How are the identity properties for multiplication The student will use problem solving, mathematical other numbers without changing the other numbers. The and addition the same? Different? For communication, mathematical reasoning, additive identity is zero (0). The multiplicative identity is each operation the identity elements are numbers that connections, and representations to one (1). There are no identity elements for subtraction combine with other numbers without changing the value  Identify a real number equation that represents each and division. of the other numbers. The additive identity is zero (0). property of operations with real numbers, when given The multiplicative identity is one (1).  The additive identity property states that the sum of several real number equations. any real number and zero is equal to the given real  What is the result of multiplying any real number by  Test the validity of properties by using examples of number (e.g., 5 + 0 = 5). zero? the properties of operations on real numbers. The product is always zero.  The multiplicative identity property states that the  Identify the property of operations with real product of any real number and one is equal to the given  Do all real numbers have a multiplicative inverse? numbers that is illustrated by a real number equation. real number (e.g., 8 · 1 = 8). No. Zero has no multiplicative inverse because there  Inverses are numbers that combine with other is no real number that can be multiplied by zero NOTE: The commutative, associative and distributive numbers and result in identity elements. resulting in a product of one. properties are taught in previous grades.  The multiplicative inverse property states that the product of a number and its multiplicative inverse (or reciprocal) always equals one (e.g., 4 · = 1).  Zero has no multiplicative inverse.  The multiplicative property of zero states that the product of any real number and zero is zero.  Division by zero is not a possible arithmetic operation. Division by zero is undefined.

6.20 The student will graph inequalities on a number line.

UNDERSTANDING THE STANDARD ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS (Background Information for Instructor Use Only)  Inequalities using the < or > symbols are  In an inequality, does the order of the elements The student will use problem solving, mathematical represented on a number line with an open circle on the matter? communication, mathematical reasoning, connections number and a shaded line over the solution set. Yes, the order does matter. For example, x > 5 is not and representation to Ex: x < 4 the same relationship as 5 > x. However, x > 5 is the  Given a simple inequality with integers, graph the same relationship as 5 < x. relationship on a number line.  Given the graph of a simple inequality with integers, represent the inequality two different ways using symbols (<, >, <, >).

 When graphing x 4 fill in the circle above the 4 to indicate that the 4 is included.  Inequalities using the ３or symbols are represented on a number line with a closed circle on the number and shaded line in the direction of the solution set.  The solution set to an inequality is the set of all numbers that make the inequality true.  It is important for students to see inequalities written with the variable before the inequality symbol and after. For example x > -6 and 7 > y.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 6 35